Restricted quantifier

A quantifier applied to predicates not with respect to the whole range of a given object variable, but with respect to a part of it defined by a predicate . When used in this restricted sense, the universal quantifier and the existential quantifier are usually denoted by and (or and , respectively). If is a predicate, then means

that is, the predicate is true for all values of the variable satisfying the predicate . The proposition means

that is, the intersection of the truth domains of the predicates and is non-empty.

Restricted quantifiers of the form and (more commonly called bounded quantifiers) play an important role in formal arithmetic (cf. Arithmetic, formal), where is a term not containing . When these quantifiers are applied to a decidable predicate, the result is a decidable predicate.